Critical state point, line, surface

From the physicochemical point of view, the prevailing one in the present review, the question is how these coefficients relate to the characteristics of the constituant molecules and how this information can be extracted from (2,3). The answer relies on the fact that the essential contributions to these integrals come from only few nonoverlapping critical regions in the joint density of states (18,19) these are points, lines and surfaces depending on the spatial extension of the conjugated electron distribution. They are defined by the condition... 

This result is similar to drained triaxial tests (namely, CTj = constant, dp = — (l/3)rfa(, dq= — da, dq/dp = 3). As shown in Fig. 6.2d, if an undrained stress path B-B under a consolidation pressure p g intersects with a drained stress path A-A" under a consolidation pressure a point C, the void ratio obtained from the undrained test is the same as the void ratio obtained from the drained test. We can draw these states in a space p, q, e) as illustrated in Fig.6.2f, which shows that the critical state is reached after travelling the surface referred to as the state boundary surface or Roscoe surface in both the undrained and drained tests. The line of failure is known as the critical state line (CSL). It is noted that the projection of CLS in the space p, q, e) onto the space (p, q) is given as = Mp, and the surface formed by CLS and its projection onto q, e) is referred to as Hvoslev surface, which is a failure surface found by Hvoslev in 1937 through a series of direct shear tests conducted on Vienna clay. 

Fig. 10. The experimentally derived surface states (thick lines) and/or surface resonances (dashed lines) and bulk band critical points (solid dots from Dongqi Li et al. 1994a open squares from Kim et al. 1992 open circles fiom Himpsel and Reihl 1983). The band structure for paramagnetic bulk gadolinium is shown with the hep symmetry labels. Some fee symmetry labels are given in brackets for comparison. (Adapted Irom Dongqi Li et al. 1994a.)...

V - Op/A. The turning point of each diagram is special there are no steady states to the left of this point two steady states exist on the right. We say that the turning point is a critical point. If we consider the family of V - aP/A diagrams traced for different values of the additional parameter T, a surface is obtained. This is a two-dimensional manifold in the three-dimensional V - oP/A - T space. As each diagram has its own turning point defined by one additional equation, the locus of critical points is a line, a one-dimensional manifold in the V - cP/A - T space. It is easy to see that the dimension of the critical manifold is na - 1, where na is the number of parameters included in the analysis. 

This figure shows schematically the P-T-composition surfaces Which represi equilibrium states of saturated vapor and saturated liquid for a binary syst The under surface represents saturated-vapor states it is the PTy surface, upper surface represents saturated-liquid states it is the PTx surface, surfaces intersect along the lines UBHCt and KAClt which represent the v pressure-vs.-T curves for pure species 1 and 2. Moreover, the under and u surfaces form a continuous rounded surface across the top of the diagram betw C and C2 the critical points of pure species 1 and 2 the critical points of... 

In these cases the critical surface can be localized relatively easily at some intersection line or surface of the electronic states involved. In addition, however, the transition probabilities must be included in (see equation (1.90)) these are different at different points of the surface. For this, the theory of transition probabilities of Landau and Zener and extended theories must be used. For details see refs. 6, 66f. 

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